3.48 \(\int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^{16}} \, dx\)

Optimal. Leaf size=84 \[ \frac{b \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{60 a^2 x^{12}}-\frac{\left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{15 a x^{15}} \]

[Out]

-((a + b*x^3)^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(15*a*x^15) + (b*(a + b*x^3)^3*
Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(60*a^2*x^12)

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Rubi [A]  time = 0.107785, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{b \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{60 a^2 x^{12}}-\frac{\left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{15 a x^{15}} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^16,x]

[Out]

-((a + b*x^3)^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(15*a*x^15) + (b*(a + b*x^3)^3*
Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(60*a^2*x^12)

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Rubi in Sympy [A]  time = 8.70683, size = 68, normalized size = 0.81 \[ - \frac{\left (2 a + 2 b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{24 a x^{15}} + \frac{\left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{60 a^{2} x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**6+2*a*b*x**3+a**2)**(3/2)/x**16,x)

[Out]

-(2*a + 2*b*x**3)*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/(24*a*x**15) + (a**2 +
2*a*b*x**3 + b**2*x**6)**(5/2)/(60*a**2*x**15)

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Mathematica [A]  time = 0.025709, size = 61, normalized size = 0.73 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (4 a^3+15 a^2 b x^3+20 a b^2 x^6+10 b^3 x^9\right )}{60 x^{15} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)/x^16,x]

[Out]

-(Sqrt[(a + b*x^3)^2]*(4*a^3 + 15*a^2*b*x^3 + 20*a*b^2*x^6 + 10*b^3*x^9))/(60*x^
15*(a + b*x^3))

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Maple [A]  time = 0.011, size = 58, normalized size = 0.7 \[ -{\frac{10\,{b}^{3}{x}^{9}+20\,a{x}^{6}{b}^{2}+15\,{x}^{3}{a}^{2}b+4\,{a}^{3}}{60\,{x}^{15} \left ( b{x}^{3}+a \right ) ^{3}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^6+2*a*b*x^3+a^2)^(3/2)/x^16,x)

[Out]

-1/60*(10*b^3*x^9+20*a*b^2*x^6+15*a^2*b*x^3+4*a^3)*((b*x^3+a)^2)^(3/2)/x^15/(b*x
^3+a)^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)/x^16,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.264416, size = 50, normalized size = 0.6 \[ -\frac{10 \, b^{3} x^{9} + 20 \, a b^{2} x^{6} + 15 \, a^{2} b x^{3} + 4 \, a^{3}}{60 \, x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)/x^16,x, algorithm="fricas")

[Out]

-1/60*(10*b^3*x^9 + 20*a*b^2*x^6 + 15*a^2*b*x^3 + 4*a^3)/x^15

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{3}{2}}}{x^{16}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**6+2*a*b*x**3+a**2)**(3/2)/x**16,x)

[Out]

Integral(((a + b*x**3)**2)**(3/2)/x**16, x)

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GIAC/XCAS [A]  time = 0.266375, size = 93, normalized size = 1.11 \[ -\frac{10 \, b^{3} x^{9}{\rm sign}\left (b x^{3} + a\right ) + 20 \, a b^{2} x^{6}{\rm sign}\left (b x^{3} + a\right ) + 15 \, a^{2} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + 4 \, a^{3}{\rm sign}\left (b x^{3} + a\right )}{60 \, x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)/x^16,x, algorithm="giac")

[Out]

-1/60*(10*b^3*x^9*sign(b*x^3 + a) + 20*a*b^2*x^6*sign(b*x^3 + a) + 15*a^2*b*x^3*
sign(b*x^3 + a) + 4*a^3*sign(b*x^3 + a))/x^15